I am told that the Falconer distance conjecture fails trivially in one dimension, but I really cannot find any reference for that. Precisely, I am asking the following question:

For a compact set $E\subseteq \mathbb R^n$, we define its distance set $\Delta(E)\subseteq [0,\infty)$ to be: $$ \Delta(E)=\{|x-y|:x,y\in E\}. $$ Then if $n=1$, we ask the following questions?

Can we find a compact set $E\subseteq [0,1]$ such that $\mathrm{dim}_H(E)>1/2$ but $\mathcal L^1(\Delta(E))=0$?

Can we find a compact set $E\subseteq [0,1]$ such that $\mathrm{dim}_H(E)=1$ but $\mathcal L^1(\Delta(E))=0$?